Two-fund separation under model mis-specification

The two-fund separation theorem tells us that an investor with quadratic
utility can separate her asset allocation decision into two steps: First,
find the tangency portfolio (TP), i.e., the portfolio of risky assets that
maximizes the Sharpe ratio (SR); and then, decide on the mix of the TP and the
risk-free asset, depending on the investor's attitude toward risk.
In this paper, we describe an extension of the two-fund separation theorem
that takes into account uncertainty in the model parameters (i.e., the
expected return vector and covariance of asset returns) and uncertainty
aversion of investors. The extension tells us that when the uncertainty
model is convex, an investor with quadratic utility and uncertainty
aversion can separate her investment problem into two steps: First, find
the portfolio of risky assets that maximizes the worst-case SR (over all
possible asset return statistics); and then, decide on the mix of this
risky portfolio and the risk-free asset, depending on the investor's attitude
toward risk. The risky portfolio is the TP corresponding to the least
favorable asset return statistics, with portfolio weights chosen optimally.
We will show that the least favorable statistics (and the associated TP)
can be found efficiently by solving a convex optimization problem.